Seismic exploration of the subterranean structure of the earth is a common practice. Typically low frequency acoustic energy is transmitted into the earth at a "shotpoint" and is detected by a "spread" of geophones; the resulting "traces", when properly organized, depict the reflection of the energy from interfaces between differing rock layers and hence provide a picture of the earth's structure.
Several methods for removal of noise to improve the signal-to-noise ratio of such low-frequency seismic signals are commonly practiced. The noise tends to be of higher frequency, such that low pass filtering methods are commonly employed. Furthermore, the noise is usually random, that is, uncorrelated. Accordingly, summing or "stacking" of "common depth point records", that is, plural records relating to reflection of the seismic energy from a single point within the earth, is useful to remove the uncorrelated noise from such low frequency data, effectively increasing the signal-to-noise ratio. These methods are well known to those of skill in the art.
Numerous types of "borehole" investigations are known in which various "logging" instruments are lowered into holes drilled into the earth. Such operations frequently generate data records in which the data takes the form of relatively high frequency impulses superimposed on low frequency background noise, which noise may be of partially correlated character, that is, generally sinusoidal. The methods of removing noise just discussed are frequently inapplicable to data records of the latter type.
For example, in borehole imaging applications, a rotating "camera" is moved axially along a borehole. The camera emits acoustic energy and detects the reflected signal. The reflected signal is indicative of the structure around the borehole, including cracks in the walls thereof. Such cracks appear as relatively high frequency impulses in the record, so that the impulses comprise the "data" to be recovered from the signal.
Copending Ser. No. 899,135, filed Aug. 21, 1986, which is assigned to the assignee of this application, and which is incorporated herein by reference, fully discusses such borehole imaging systems. As discussed therein, noise in such signals tends to be low frequency noise caused by departure of the rotating camera from the center of the borehole. As noted, the data to be recovered is "impulses", that is, relatively discontinuous high frequency changes in the amplitude of the reflected signal. The data impulses correspond to fractures in the earth's structure surrounding the borehole. Since these fractures indicate possible locations for removal of hydrocarbons, and provide additional clues to the structure of interest, the impulses in the signal are very significant. To interpret such a signal accurately requires that the "impulses" be effectively removed from the noise. As indicated, the noise typically relates to the spacing of the tool from the center of the borehole, and other generally correlated variables, so that the noise tends to be of sinusoidal, slowly varying character as the tool moves from one part of the hole to another.
High frequency "impulse" data which is superimposed on and obscured by low frequency noise is also encountered in connection with the so-called "casing collar locating" technique. As will be understood by those of skill in the art, "casing collars" are pipe couplings which connect the sections of pipe making up the casing of a borehole. These collars can be used to provide absolute indications of the depth of the corresponding logging tool in the hole, if they can be reliably detected. Other methods of measurement of depth tend to be inaccurate, due to cable stretch and other factors. Accordingly, it is conventional to use the location of the collars in the casings as objective indications of depth.
A typical casing collar locating tool comprises a magnetic device for detecting variations in the amount of ferromagnetic material in its vicinity, i.e. the steel of the pipe and collar. When such a tool is moved up and down in the borehole, a low frequency signal with impulses is generated. The overall amplitude of the signal varies slowly, again due to variation in spacing of the tool from the center of the hole and other slowly-varying parameters, while relatively high frequency impulses are superimposed thereon corresponding to the casing collars. As indicated above, it would be desirable to remove the low frequency noise, leaving the impulses indicating the presence of the collars, such that the tool depth data thus collected could be adequately analyzed.
Prior techniques for removing low frequency noise and leaving the high frequency data behind have not been as efficient as desired. One possibility would clearly be high pass filtering. Specifically, if the received data signal Z(t) is equal to the sum of the actual signal S(t) and the noise signal N(t), and if S(t) and N(t) have clearly differing frequencies, filtering can be relatively efficient in removing N(t). However, if as is more common the combination is multiplicative, such that Z(t)=S(t)N(t), simple filtering is not adequate, and normally involves some distortion of the signal.
One attempt to solve this problem is described in U.S. patent application Ser. No. 899,135 referred to above. In that application a "homomorphic" filtering technique is described. According to this technique, the logarithm of Z(t) is taken, and 1n Z(t) is then decomposed by filtering. If Z(t)=S(t)N(t), as above, 1n Z(t)=1n S(t)+1n N(t). Hence if the frequencies of N(t) and S(t) differ, frequency domain filtering can be employed to separate N(t) from S(t). The result can then be transformed back to the time domain and exponentiated, yielding S(t).
However, unless N(t) and S(t) are indeed primarily located in different frequency bands, the homomorphic filtering technique poses certain problems, particularly in connection with the selection of the appropriate cutoff frequency. In general, any linear filtering technique tends to distort the data; this makes it difficult, for example, to interpret the casing collar log data accurately.
Accordingly there is a need in the art for an improved method of processing seismic data of the type in which the data takes the form of impulses superimposed on relatively low frequency, partially correlated noise.
The present inventor is aware of a conventional digital signal processing technique known as "median filtering", which has been employed in connection with processing of television signals and the like. Median filtering is described a number of references, including the following:
T. S. Huang, ed., Two Dimensional Signal Processing II: Topics in Applied Physics, Vol. 43, Chap. 5 & 6, Springer Verlag, Berlin, 1981; N. C. Gallagher and G. L. Wise, "A Theoretical Analysis of the Properties of Median Filters", IEEE Trans. on ASSP, vol. ASSP-29, No. 6, December, 1981; N. S. Jayant, "Average and Median Based Smoothing Techniques for Improving Digital Speech Quality in the Presence of Transmission Errors", IEEE Trans. on Communications, vol. COM-24, pp. 1043-1045, September, 1976; T. S. Huang, G. L. Yang and G. Y. Yang, "A Fast Two-Dimensional Median Filtering Algorithm", IEEE Trans. on ASSP, vol. ASSP-27, pp. 13-18, February, 1979; G. R. Arce and N. C. Gallagher, "Stochastic Analysis for the Multiband Roof Signal Set of Median Filters", 16th Annual Princeton Conf. on Information Sciences and Systems, March 1982; and A. C. Bovik, T. S. Huang and D. C. Munson Jr., "A Generalization of Median Filtering Using Linear Combination of Order Statistics", IEEE Trans. on ASSP, vol. ASSP-31, No. 6, December, 1983, pp. 1342-1349.
In median filtering, successive subsets of a complete series of samples of the signal are placed into successive "windows" containing an odd number M of samples. For example, if M=5, the first window of samples S.sub.1, S.sub.2 . . . S.sub.n would contain S.sub.1, S.sub.2, S.sub.3, S.sub.4 and S.sub.5 ; the second S.sub.2 -S.sub.6 ; and so on. The samples in each window are then ordered by magnitude. The median value, that is, the value which is in the center of the ordered samples, then becomes the output sample corresponding to each window. That is, the median value of the data samples in each window becomes the corresponding value of the filtered signal.
For example, suppose the signals in a given seven-sample window are 1, 11, 8, 3, 4, 6, and 2. Conventional linear filtering would take the average of that window, that is, add up the values and divide by the number of samples, in this case 35/7=5. According to the median filtering technique, the samples are first ordered, thus appearing as 1, 2, 3, 4, 6, 8, 11. The median or central value, in this case 4, is then selected as the value of the filtered signal corresponding to that particular window. The window is then moved over one sample, and the process repeated.
Because the samples of the higher amplitude impulse data (in the example, the 8 and 11 values) will be located to one end of each window after the samples are ordered, median filtering can be used effectively to remove impulses from signals. That is, since the samples of the impulse data are always at one end of the window, they never become the median value in the ordered data in the window, and are never selected as the value for the filtered data signal. Accordingly, the impulses are removed.
The degree to which median filtering removes impulses depends in a simple and intuitively understood manner on the length of the window relative to the width of the impulses. The median-filtering process is well understood, as discussed in the references listed above, and can readily be implemented to remove impulses as desired.